Uniqueness in determining refractive indices by formally determined far-field data

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1 Applicable Analysis, 2015 Vol. 94, No. 6, , Uniqueness in determining refractive indices by formally determined far-field data Guanghui Hu a, Jingzhi Li b and Hongyu Liu c a Weierstrass Institute (WIAS), Mohrenstr. 39, Berlin 10117, Germany; b Faculty of Science, South University of Science and Technology of China, Shenzhen , P.R. China; c Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR Communicated by M. Klibanov (Received 3 April 2014; accepted 3 May 2014) We present two uniqueness results for the inverse problem of determining an index of refraction by the corresponding acoustic far-field measurement encoded into the scattering amplitude. The first one is a local uniqueness in determining a variable index of refraction by the fixed incident-direction scattering amplitude. The inverse problem is formally posed with such measurement data. The second one is a global uniqueness in determining a constant refractive index by a single far-field measurement. The arguments are based on the study of certain non-selfadjoint interior transmission eigenvalue problems. Keywords: inverse scattering; refractive index; uniqueness; formally-determined; interior transmission eigenvalues AMS Subject Classifications: 35A02; 35R30 1. Introduction This note is concerned with the inverse acoustic scattering problem of recovering the refractive index of an inhomogeneous medium. Suppose in the homogeneous space, there is an inhomogeneity. In order to determine the inhomogeneity, one sends a certain detecting wave field. The propagation of the detecting wave field will be perturbed when meeting the inhomogeneity. The perturbation is the so-called scattering in the literature. The inverse problem is to recover the inhomogeneity by measuring the corresponding scattering wave field. The study of inverse scattering problems lies in the core of many areas of science and technology, such as radar and sonar, geophysical exploration, medical imaging, nondestructive testing, and remote sensing; see [1 6] and the references therein. Throughout, we shall take the incident field to be the time-harmonic acoustic plane wave of the form u i (x) := e ikx d, x R N with N 2, where k R + is the wave number and d S N 1 is the impinging direction. The optical property of the homogeneous space is described as an index of refraction n(x), which is normalized to be 1, whereas in an open domain accommodating the inhomogeneity, it is assumed that n(x) 1 \0, x.we Corresponding author. li.jz@sustc.edu.cn 2014 Taylor & Francis

2 1260 G. Hu et al. assume that is a bounded Lipschitz domain with R N \ connected, and that the refractive index n(x) L () satisfying n 1 L () ɛ 0 > 0. The wave propagation is governed by the following Helmholtz equation u(x) + k 2 n 2 (x)u(x) = 0, x R N, (1.1) where u(x) denotes the wave pressure. We seek a solution u Hloc 1 (RN ) such that u(x) = u i (x) + u s (x), x R N \. (1.2) u s (x) is called the scattered wave field, which satisfies the so-called Sommerfeld radiation condition, { N 1 u s } (x) lim x 2 iku s (x) = 0. (1.3) x x The well-posedness of the scattering system (1.1) (1.3) is well understood (cf. [4,7]). Particularly, u s admits the following asymptotic expansion ( ) u s (x) = eik x 1 u x N 1 ( ˆx; d, k) + O, x +, (1.4) 2 x N 2 which holds uniformly in all directions ˆx := x/ x, x R N. u ( ˆx; d, k) is known as the scattering amplitude. By the celebrated Rellich s Lemma (cf. [3]), the scattering amplitude u ( ˆx) encodes all the information of the scattered wave field u s (x). The inverse scattering problem that we consider in the present study is to recover the scatterer (, n) by knowledge of u ( ˆx; d, k). If one introduces an operator F which sends the scatterer (, n) to the corresponding scattering amplitude, then the inverse scattering problem can be abstractly formulated as the following operator equation F(n) = u ( ˆx; d, k). (1.5) It is easy to verify that the operator Equation (1.5) is non-linear and also widely known to be ill-posed (cf. [3,4]). One of the foundational issues in the inverse scattering problem is the uniqueness/ identifiability: can one really identify the scatterer by the measurement? how many measurement data one should use for the identification? Mathematically, the uniqueness issue can be stated as follows. Let n 1 and n 2 be two refractive indices with the scattering amplitudes u 1 ( ˆx; d, k) and u2 ( ˆx; d, k). Then u 1 ( ˆx; d, k) = u2 ( ˆx; d, k) iff n 1 = n 2. (1.6) The study on uniqueness is usually very difficult and challenging. Intensive efforts have been devoted to the unique determination of many inverse scattering problems and quite different technical treatments and mathematical theories are needed with different problems. Some significant developments have been achieved for several representative inverse scattering problems; see [1,3,4,6,8 11] and the references therein. One uniqueness result for the inverse problem (1.5) is due to Sylvester and Uhlmann (cf. [12]; see also [3,4,13]). It is shown that in R N, N 3, u ( ˆx; d, k) with a fixed k R + and all ( ˆx, d) S N 1 S N 1 uniquely determines n(x). Here, we note that the measurement data used are over-determined. Indeed, one can easily count that the dimensions of the known (namely u ( ˆx; d, k)) and the unknown (namely n(x)), are respectively, 2N 2 and N. Bydimension, we mean the

3 Applicable Analysis 1261 number of independent variables in a set. Clearly, one has 2N 2 > N if N 3. This motivates us to consider the unique determination of the refractive index function n(x) by u ( ˆx; d, k) with a fixed d S N 1 and all ( ˆx, k) S N 1 R +. It is easily seen that the inverse scattering problem is formally posed with such measurement data. Since u ( ˆx; d, k) is (real) analytic with all its arguments and due to analytic continuation, the measurement data-set could be replaced by u ( ˆx; d, k) for a fixed d S N 1, and all ( ˆx, k), where is any open subset of S N 1 R +. To our best knowledge, there is no uniqueness result in the literature by using such a formally determined data-set. Our study connects to that of the non-self-adjoint interior transmission eigenvalue problems, which was first introduced by Colton and Monk in [14], and were recently extensively studied; see [15 18] and the references therein. We conjecture that the uniqueness can be established in a very generic setting by using the formally determined data-set. However, in Section 2, we shall only present a local uniqueness result in determining a variable refractive index, by directly implementing the discreteness of interior transmission eigenvalues due to Sylvester in [18]. Our main contribution in Section 2 is to bridge the study on uniqueness of the inverse scattering problem and that on the interior transmission eigenvalue problems, which in our hope might open up some new directions of research in the relevant field. In Section 3, a much more interesting uniqueness result is derived in determining a constant refractive index by a single far-field measurement, namely u ( ˆx; d, k) with both d and k fixed, and all ˆx S N 1. The uniqueness by a single far-field measurement is extremely challenging for inverse scattering problems in the literature; see Section 4 for more discussion on this aspect. 2. Local uniqueness in determining a variable refractive index Throughout, we let n and n be two positive constants such that n < n.let(, n j ), j = 1, 2, be two inhomogeneous media as described in Section 1 satisfying n n j (x) n for x. The scattering problem (1.1) (1.3) corresponding to (, n j ) can be easily formulated as the following transmission eigenvalue problem: find a pair of solutions (u j, u s j ) H 1 () Hloc 1 (RN \) such that u j + k 2 n 2 j u j = 0 in, u s j + k2 u s j = 0 in RN \, u j = u s j + u j ui, ν = us j ν + ui on, (2.1) { ν} N 1 u s j (x) lim x 2 iku s j x x (x) = 0, where ν denotes the exterior unit normal vector to.weletu j ( ˆx; d, k) be the scattering amplitude of (2.1). Suppose that for a fixed d S N 1 and a fixed k R +, Then by Rellich s Lemma (cf. [3]), we have u 1 ( ˆx; d, k) = u2 ( ˆx; d, k) for ˆx SN 1. (2.2) u s 1 (x) = us 2 (x), x RN \, (2.3)

4 1262 G. Hu et al. By (2.1) and (2.3), it is easy to see that (u 1, u 2 ) H 1 () H 1 () satisfies u 1 + k 2 n 2 1 u 1 = 0 in, u 2 + k 2 n 2 2 u 2 = 0 in, u 1 u 1 = u 2, ν = u 2 on. ν (2.4) Next, we show that (u 1, u 2 ) must be non-trivial solutions. It is easily seen that if either one of u 1 and u 2 is a trivial solution, then both of them are trivial solutions to (2.4). Without loss of generality, we suppose that u 1 = 0. Then by (2.1), we have that u i + u s 1 = ν(u i + u s 1 ) = 0 on. Hence, by Holmgren s Theorem (cf. [3]), one must have that u i + u s 1 = 0inRN \, which contradicts with the fact that lim x us 1 (x) + eikx d =1. Hence, (u 1, u 2 ) is a pair of non-trivial solutions to (2.4). According to [14], k is an interior transmission eigenvalue to (2.4) with u 1 and u 2 being the corresponding eigenfunctions. Based on the above observation, one can show the uniqueness (1.6) with a fixed d S N 1 and all ( ˆx, k) S N 1 R + by absurdity as follows. If u 1 ( ˆx; d, k) = u2 ( ˆx; d, k) for all ( ˆx, k) S N 1 R +, then by a similar argument to derive (2.4), we know every k R + is an interior transmission eigenvalue with u 1 (x; d, k) and u 2 (x; d, k) being the corresponding eigenfunctions. If one can show that the interior transmission eigenvalues to (2.4) for certain admissible n 1 and n 2 are discrete, then one obviously arrives at a contradiction. However, such discreteness of the interior transmission eigenvalues for (2.4) is only available for certain restricted n 1 and n 2 in the literature. The connection discussed above obviously bridges the study on uniqueness of the inverse scattering problem and that on the interior transmission eigenvalue problems, which in our hope might open up some new directions of research in the relevant field. Next, we present a local uniqueness result in determining a refractive index by using the fixed-incident-direction scattering amplitude. Theorem 2.1 Let (, n) be an inhomogeneous medium with n < n(x) <n for x and let ε(x) L (). Let u and v denote the scattering amplitudes corresponding to (, n) and (, n + ε), respectively. If there exisit two positive constants ɛ + and ɛ, and a neighborhood of,neigh( ), such that ɛ + <ɛ < n, ε(x) ɛ for x, and either ε(x) ɛ + or ε(x) ɛ + for x neigh( ).Then one cannot have that for any fixed d S N 1, and all ( ˆx, k) S N 1 R +. u ( ˆx; d, k) = v ( ˆx; d, k) (2.5) Proof Assume that (2.5) holds for a fixed d S N 1 and all ( ˆx, k) S N 1 R +. Then, according to our earlier discussion, we know that every k R + is an interior transmission eigenvalue to

5 Applicable Analysis 1263 u + k 2 n 2 u = 0 v + k 2 (n + ε) 2 v = 0 u u = v, ν = v ν in, in, on. (2.6) However, in [18], it is shown that provided n and ε satisfy the assumptions stated in the theorem, the interior transmission eigenvalue problem (2.6) possesses (at most) a discrete set of eigenvalues, which immediately yields a contradiction. The proof is completed. 3. Uniqueness in determining a constant refractive index In this section, we shall prove that an inhomogeneous medium with a constant refractive index can be uniquely determined by a single far-field measurement. Indeed, we have Theorem 3.1 Let (, n) be an inhomogeneous medium with the refractive index n being a complex-valued constant such that In 0 and 0 < n n, and let u ( ˆx; d, k) be the associated scattering amplitude. Then there exists a positive constant k 0, depending only on n and, such that n is uniquely determined by u ( ˆx; d, k) with any fixed 0 < k < k 0, d S N 1, and all ˆx S N 1. Remark 3.1 In Theorem 3.1, we allow n to be complex-valued. In is known as absorbing or damping coefficient of the acoustic medium (, n). Proof As before, we shall prove the theorem by absurdity. Let (, ñ) be another inhomogeneous medium with the refractive index ñ being a constant such that Iñ 0, 0 < ñ n, ñ = n and u ( ˆx; d, k) = v ( ˆx; d, k) for fixed k > 0 and d S N 1, and all ˆx S N 1, (3.1) where v ( ˆx; d, k) denote the scattering amplitude corresponding to (, ñ). By a similar argument to that in deriving (2.4), we know (u,v) H 1 () H 1 () is a pair of interior transmission eigenfunctions to the following system u + k 2 n 2 u = 0 in, v + k 2 ñ 2 v = 0 in, u u = v, ν = v (3.2) on. ν Set u v w := k 2 (ñ 2 n 2 ). It is straightforward to show that (w, u) H 1 () H 1 () satisfy w + k 2 n 2 w = v in, v + k 2 ñ 2 v = 0 w w = 0, ν = 0 in, on. (3.3)

6 1264 G. Hu et al. Next, multiplying both sides of the first equation in (3.3) byv and then integrating over, we have v 2 dx = (w + k 2 n 2 w) v dx (3.4) = (w + k 2 n 2 w) v dx (v + k 2 ñ 2 v) w dx (3.5) ( w = ν v v ) ν w ds(x) + k 2 (n 2 ñ 2 ) v w dx (3.6) = k 2 (n 2 ñ 2 ) v w dx. (3.7) From (3.4) to(3.5), we have made use of the second equation in (3.3); from (3.5) to(3.6), we have made use of Green s formula; and from (3.6) to(3.7), we have made use of the homogeneous boundary conditions for w in (3.3). Now, by (3.4) (3.7), we clearly have v 2 dx 2k 2 n 2 v L 2 () w L 2 (). (3.8) By Lemma 3.1 in the following, we know that provided k 0 is sufficiently small, w L 2 () C() v L 2 (), (3.9) where C() is a positive constant depending only on. Finally, we further require that k 0 < 1 2C() n. (3.10) Then, by (3.8) (3.10), it is straightforward to show that v L 2 () = 0, which immediately yields a contradiction. The proof is completed. Lemma 3.1 Let f L 2 () and w H 1 () satisfy w + k 2 w = f in, w = 0 on. (3.11) Then there exist positive constants k 0 = k 0 () and C = C() such that w L 2 () C() f L 2 (), for all k < k 0(). (3.12) Proof By integrating by parts, we first have f w dx = (w + k 2 w) w dx = w 2 dx + k 2 w 2 dx. (3.13) Next, by Poincaré inequality, we have w 2 dx C 1 () w 2 dx, (3.14)

7 Applicable Analysis 1265 where C 1 () is a positive constant depending only on. Then, by (3.13) and (3.14), we further have 1 w 2 dx w 2 dx k 2 w 2 dx + f w dx C 1 () k 2 w 2 dx + α w 2 dx + 1 f 2 dx 4α k0 2 w 2 dx + α w 2 dx + 1 f 2 dx, (3.15) 4α where α R +. By choosing 1 k 0 = 2 (3.16) C 1 () and letting α = k0 2 in (3.15), we can compute directly that w 2 dx 2C 1 () 2 f 2 dx. (3.17) Therefore, the lemma is proved by taking C() = 2C 1 (). In the rest of this section, we present a uniqueness result in determining a spherically symmetric refractive index by a single far-field measurement without the smallness condition on k in Theorem 3.1. Theorem 3.2 Let (, n) be an inhomogeneous medium with a constant refractive index n = 0. Suppose further that = B R (z) := {x R N : x z =R} is a ball centered at z R N with radius R > 0. Then, the ball (that is, R and z) and its refractive index n can be uniquely determined by the far-field pattern u ( ˆx; d, k) for all ˆx S N 1 with any fixed incident direction d S N 1 and wave number k R +. Proof We carry out the proof following the argument in [3, Theorem 5.4] for the unique determination of sound-soft balls. Let (, ñ) be another spherically symmetric medium with the constant refractive index ñ, where = B R ( z). Denote by ũ ( ˆx; d, k) the farfield pattern corresponding to incited by the plane wave u i = exp(ikx d). Assuming ũ ( ˆx; d, k) = u ( ˆx; d, k) for all ˆx S N 1 (N = 2, 3), we shall prove z = z, R = R and n =ñ. By Rellich s Lemma, the scattered waves u s (x; B R (z)) and ũ s (x, B R ( z)) coincide in R N \B R (z) B R ( z). We first prove z = z, i.e. the centers of and coincide. Similar to the proof of [3, Theorem 5.4], we claim that the scattered field u s (x; B R (z)) can be analytically extended from R 3 \ into R 3 \{z}. In fact, this can be derived from the explicit expression of u s in terms of k, n, R and z. Without loss of generality we suppose N = 3. We make an ansatz on the scattered field u s (x; B R (z)) and the transmitted wave u(x; B R (z)) that u s (x; B R (z)) = u(x; B R (z)) = m=0 i m (2m + 1) A m h (1) m (k x z )P m(cos θ), A m C, x > R, (3.18) i m (2m + 1) B m j m (kn x z )P m (cos θ), B m C, x < R, m=0

8 1266 G. Hu et al. where h m (1) denotes the Hankel function of the first kind of order m, j m the Bessel functions of order m, P m the Legendre polynomials and θ the angle between d and x z. Recalling the Jacobi-Anger expansion (see e.g. [3, (2.46)]) e ikx d = e ikz d e ik(x z) d = e ikz d m=0 i m (2m + 1) j m (k x z ) P m (cos θ), x R 3, and taking into account the transmission conditions between u and u s on x =R, we obtain the following algebraic equations for A m and B m : ( ) ( h m (1) ) ( ) (t) j m (tn) Am th m (1) (t) tn j m (tn) = e ikz d jm (t) B m tj m (t), t := kr. Simple calculations show that A n = e ikz d j m (t) j m(tn) nj m (t) j m (tn) h m (1) (t) j m (tn) nh m (1) (t) j m (tn). By the asymptotic behavior of the spherical Bessel and Hankel functions as n and their differential formulas (see, e.g. [3, Chapter 2.4]), we have t m ( ( )) ( ( )) 1 j m (t) = 1 + O, j ntm 1 1 m (t) = 1 + O, (2m + 1)!! m (2m + 1)!! m ( ( )) ( ( )) h m (1) (2m 1)!! 1 (t) = it m O, h m (1) (m + 1)(2m 1)!! 1 (t) = m it m O, m where m!! = m for any odd number m N +. Inserting the previous asympotics into the expression of A n leads to the estimate ( A m = O t 2m+1 (2m + 1)!! (2m + 1)!! ) as m, and thus ( (2m + 1)A m h m (1) (k x z ) = O R 2m+1 k m ) x z m+1. (2m + 1)!! This implies that the scattered field u s (x, B R (z)) converges uniformly in any compact subset of R 3 \{z}. Analogously, ũ s (x, B R ( z)) has an extension from R3 \B R ( z) into R3 \{ z}. Now, we assume z = z. Defining u s (x, B R (z)) x=z := ũ s (z, B R ( z)), we obtain an entire function u s (x, B R (z)) that satisfies the Helmholtz equation and the Sommerfeld radiation condition, leading to u s (x, B R (z)) = 0inR 3. This contradiction indicate that one must have z = z. From the series (3.18), one readily concludes that u and ũ depend only on the angle θ between the incident direction d and the observation direction (x z)/ x z. Hence, the relation ũ ( ˆx; d, k) = u ( ˆx; d, k) for one incident direction implies the coincidence of far-field patterns for all incident directions. Consequently, we have R = R and n =ñ due to the uniqueness in the inverse medium problem using all incident and observation directions; see e.g. [3, Chapter 10.2] where the refractive index is allowed to be piecewise continuous. This completes the proof in 3D. For the two-dimensional case, the determination of the center of the disc B R (z) can be shown in a completely manner to the 3D case. As soon as the center of B R (z) is recovered,

9 Applicable Analysis 1267 the uniqueness in determining R and n directly follows from the uniqueness in determining a potential for the 2D Schrödinger equation in [19];seealso[20] and [21]. The proof is completed. By the invariance of the Helmholtz equation under rotations, it is easily verified that for an inhomogeneous medium (B R (z), n( x z )), knowing the scattering amplitude for a fixed incident direction is equivalent to knowing the scattering amplitude for all incident directions. Hence, by a completely similar argument to the proof of Theorem 3.2, we have Theorem 3.3 Let := B R (z) be a ball of radius R > 0 centered at z in R N.Let(, n) be an inhomogeneous medium with the refractive index n := n( x z ) L () being a complex-valued function of x z. It is supposed that the center z is known in advance. Then the refractive index n can be uniquely determined by the associated scattering amplitude u ( ˆx; d, k) with any fixed k R +,d S N 1 and all ˆx S N Concluding remarks One of the classical inverse scattering problems is the so-called obstacle problem.letd be a bounded Lipschitz domain such that R N \D is connected. Consider the following scattering problem u + k 2 u = 0 in R N \D, u = u i + u s in R N \D, u = 0 on D, (4.1) } lim x x N 1 2 iku s (x) = 0. { u s (x) x The scattered field in (4.1) possesses the same asymptotic expansion as that in (1.4). The inverse obstacle scattering problem is to recover D by knowledge of u ( ˆx; d, k). D is known as a sound-soft obstacle in the physical literature. It has been long conjectured that D can be uniquely determined by a single far-field measurement, namely u ( ˆx; d, k) for all ˆx S N 1, but fixed k R + and d S N 1. We note that the inverse obstacle problem is formally posed by a single far-field measurement. The first uniqueness result is due to Schiffer (cf. [22]), where the uniqueness was established by infinitely many far-field measurements, namely u ( ˆx; d, k) with all ˆx S N 1, and either i). a fixed d S N 1 and infinitely many k s; or ii). a fixed k R + and infinitely many d s. The corresponding proof is based on an absurdity argument, and the essential ingredient is the discreteness of the Dirichlet eigenvalues for the negative Laplacian in a bounded domain. In this sense, our argument in Section 2 on uniquely determining a generic inhomogeneous medium can be taken as a counterpart to Schiffer s argument on uniquely determining a generic sound-soft obstacle. However, in order to establish the uniqueness of determining a generic inhomogeneous medium, one has to deal with the more challenging interior transmission eigenvalue problems (2.4). We conjecture that a generic refractive index n(x) can be uniquely determined the formally-determined fixed-incident-direction scattering amplitude. A possible way to achieve this goal is to study the discreteness of the interior transmission eigenvalues of (2.4), but one needs to peel off the singular regions where n 1 = n 2 from exterior. We shall present such a study in the future. Moreover, it is interesting to point out that our proposed inverse scattering problem is closely connected to the coefficient inverse problems by a single dynamical measurement; see [8,9,23 26].

10 1268 G. Hu et al. By using the fact that there exists a lower bound on the Dirichlet eigenvalues, Colton and Sleeman established the uniqueness with a single far-field measurement for the inverse obstacle problem provided the obstacle is sufficiently small (see [27]). Our uniqueness in Section 3 on uniquely determining a constant refractive index by a single far-field measurement can be taken as a counterpart to that uniqueness due to Colton and Sleeman. In order to prove Theorem 3.1, we actually have shown that there is a lower bound for the positive interior transmission eigenvalues of (3.1). The lower bound of positive interior transmission eigenvalues was also considered in [28]. However, our approach works in a more general setting than that was considered in [28]. Finally, we note that there are some significant progresses in uniquely determining a generic obstacle by using a single farfield measurement even without the smallness condition; see [10,29 31] and the references therein. We believe that a generic constant refractive index can also be uniquely determined by a single far-field measurement without the smallness condition posed in Theorem 3.1. Indeed, Theorems 3.2 and 3.3 cast some light on this conjecture. Funding This work was supported by the German Research Foundation (DFG), [grant number HU 2111/1-1]; the NSF of China [grant number ], [grant number ], [grant number ] and the NSF grant, DMS References [1] Ammari H, Kang H. Reconstruction of small inhomogeneities from boundary measurements. Vol. 1846, Lecture notes in mathematics. Berlin: Springer-Verlag; [2] Ammari H, Kang H. Polarization and moment tensors. With applications to inverse problems and effective medium theory. New York (NY): Applied Mathematical Sciences, Springer; [3] Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory. 2nd ed. New York (NY): Springer; [4] Isakov V. Inverse problems for partial differential equations. 2nd ed. Vol. 127, Applied mathematical sciences. New York (NY): Springer-Verlag; [5] Pike R, Sabatier P, editors. Scattering: scattering and inverse scattering in pure and applied science. San Diego (CA): Academic Press; [6] Uhlmann G, editor. Inside out: inverse problems and applications, MSRI publications. Vol. 47. Cambridge: Cambridge University Press; [7] Liu HY, Shang Z, Sun H, Zou J. Singular perturbation of reduced wave equation and scattering from an embedded obstacle. J. Dyn. Differ. Equ. 2012;24: [8] Bukhgeim AL, Klibanov MV. Uniqueness in the large of a class of multidimensional inverse problems. Soviet Math. Doklady. 1981;17: [9] Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-posed Probl. 2013;21: [10] Liu HY, Zou J. On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems. J. Phys. Conf. Series. 2008;124: [11] Uhlmann G. Developments in inverse problems since Calderón s foundational paper, Chapter 19. In: Christ M, Kenig C, Sadosky C, editors. Harmonic analysis and partial differential equations. London: University of Chicago Press; p [12] Sylvester J, Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2). 1987;125:

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